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Results tagged with formal-schemes
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user 127776
2
votes
0
answers
162
views
Algebraization of vector bundles over non-algebraically closed fields
I've asked this question here but never got an answer, a simplified version of the question is the following:
Given an intersection of hypersurfaces $Y$ on a smooth projective variety $X$ over a finit …
- 5,901
4
votes
0
answers
143
views
On formal completions of normal bundles in the non-affine case
According to this post. In the affine case the formal completion of $A$ along an ideal $I$ coincides with the formal is isomorphic to the completion of the normal bundle along its zero section. I was …
- 5,901
5
votes
0
answers
253
views
Colimit of nilpotent thickenings in the category of schemes
This question is highly related to this and this one.
Given a ring $A$ and an ideal $I$, the direct system of schemes $\text{Spec}(A/I)\rightarrow \text{Spec}(A/I^2)\rightarrow \ldots$ has a colimit i …
- 5,901
4
votes
0
answers
277
views
How much does the formal completion know about the ambient variety?
How much information does the formal completion along an ample divisor hold about the ambient variety? Given two smooth projective varieties such that they have isomorphic ample divisors, where the fo …
- 5,901
4
votes
1
answer
333
views
Symmetric powers of curves and completion along the diagonal
Given a smooth curve $C$, denote by $\text{Sym}^d(C)$ its $d$-th symmetric power. Let $\Delta$ be the diagonal subvariety which is defined as the codimension $1$ subvariety that at least two of the po …
- 5,901
12
votes
0
answers
582
views
Global version of Gabber's rigidity theorem
I had a question regarding Gabber's rigidity.
Let $A$ be a ring (let's assume Noetherian) and $I$ be an ideal, since the pair $(\hat{A},I)$ is a henselian pair ($\hat{A}$ is the completion along $I$), …
- 5,901
2
votes
0
answers
149
views
Examples of effective Lefschetz condition
When a closed subvariety $Y$ of $X$ satisfy effective Lefschetz condition $Leff(X,Y)$, it implies there is an equivalence if categories between the vector bundles on the formal neighborhood of $Y$ in …
- 5,901
4
votes
1
answer
375
views
Vector bundles on complete rings
Given a ring $A$ and an ideal $I$, consider the completion $\hat{A}$. What does usually mean by a vector bundle on $\hat{A}$? One way is to consider projective $\hat{A}$-modules. Another one is a syst …
- 5,901
2
votes
0
answers
150
views
Colimits of the infinitesimal neighborhoods of symmetric product in the category of schemes
This problem is highly related to this one and in fact it is the same question applied to a very specific situation.
Given a smooth projective curve $C$, let $\text{Sym}^i(C)$ be the $i$-th symmetric …
- 5,901
3
votes
0
answers
238
views
A question about extending vector bundles from formal neighborhood to a coherent sheaf
I have a question which probably is very straightforward but because of my lack of knowledge of formal schemes I'm asking it here. Let's assume we have a vector bundle $E$ on the formal completion $X_ …
- 5,901
2
votes
0
answers
259
views
Zariski descent of algebraic $K$-theory on formal schemes
This question is highly related to some other questions that I've previously asked, especially to this one. In this problem we have a scheme $X$ and a closed subscheme $Z$ the formal completion $X_Z$. …
- 5,901